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Grothendieck–Plücker Images of Hilbert Schemes are Degenerate

Part of: Computational aspects and applications Cycles and subschemes

Published online by Cambridge University Press:  22 August 2018

Donghoon Hyeon  and
Hyungju Park
Donghoon Hyeon
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul, Republic of Korea (dhyeon@snu.ac.kr; parkhyoungju@snu.ac.kr)
Hyungju Park
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul, Republic of Korea (dhyeon@snu.ac.kr; parkhyoungju@snu.ac.kr)
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Abstract

We study the decompositions of Hilbert schemes induced by the Schubert cell decomposition of the Grassmannian variety and show that Hilbert schemes admit a stratification into locally closed subschemes along which the generic initial ideals remain the same. We give two applications. First, we give completely geometric proofs of the existence of the generic initial ideals and of their Borel fixed properties. Second, we prove that when a Hilbert scheme of non-constant Hilbert polynomial is embedded by the Grothendieck–Plücker embedding of a high enough degree, it must be degenerate.

Keywords

generic initial idealsSchubert cell decomposition

MSC classification

Primary: 14C05: Parametrization (Chow and Hilbert schemes)
Secondary: 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 47 - 60
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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